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Radius of Curvature is the approximate radius of a circle at any point. The radius of curvature changes or modifies as we move further along the curve. The radius of curvature is denoted by R. Curvature is the amount by which a curved shape derives from a plane to a curve and from a bend back to a line. It is a scalar quantity. The radius of curvature is basically the reciprocal of curvature.
The radius of curvature can be calculated for any curve with the equation y = f(x) with x as its parameter. The Radius of Curvature Formula is given as R =(1 + (dy/dx)2)3/2 / |d2y/dx2|, where, dy/dx refers to the first derivative of the function y = f(x), and d2y/dx2 refers to the second derivative of the function y = f(x).
Read More: NCERT Solutions for Class 10 Maths Circles
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Key Terms: Radius of Curvature, Radius of Curvature Formula, Radius, Circle, Curve, Derivative, Extrinsic Curvature, Intrinsic Curvature
Radius of Curvature Formula
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Radius of Curvature is the reciprocal of the curvature in differential geometry. Any approximate radius of a circle at any given point is called the radius of curvature of the curve. The vector length of curvature is also called the radius of curvature.
Radius of Curvature
For any curve with the equation, y = f(x) with x as a parameter, the Radius of Curvature Formula can be given as
R=(1+(dy/dx)2)3/2/|d2y/dx2|
Where
- dy/dx: First derivative of the function y = f(x),
- d2y/dx2: Second derivative of the function y = f(x).
Radius of Curvature Formula is also given as
R = 1/K
Where R denotes the length or radius of curvature and K is the derivative of curvature.
Radius of Curvature Formula
Read More:
| Relevant Topics | ||
|---|---|---|
| Areas Related to Circles | Area of a Circle | Standard Equation of a Circle |
| Tangent to a Circle | The Sector of a Circle | Areas Related To Circles Formula |
Solved Examples on Radius of Curvature Formula
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Given below are a few solved examples of the Radius of Curvature Formula to understand the concept better:
Example 1: Find the radius of curvature for f(x) = 4x2 + 3x – 7 at x = 4.
Solution: We have y = 4x2 + 3x - 7 and x = 4.
- dy/dx = 16x + 3
- d2y/dx2 = 16
Using the radius of curvature formula,
we get R=(1+(dy/dx)2)3/2/|d2y/dx2|
= (1 + 256x2 + 9 + 36x)3/2/16
= (256x2 + 36x + 10)3/2/16
Substitute the value x = 4.
R = (256 (4)2 + 36 (4) + 10 )3/2/16
= (4096 + 144 + 10)3/2/16
= 27066/16
= 1691.625 units
Example 2: Find the radius of curvature for f(x) = 3x2 + 3x - 2 at x = 1.
Solution: We got y = 3x2 + 3x - 2 and x = 1.
- dy/dx = 6x + 3
- d2y/dx2 = 6
Using the radius of curvature formula,
R=(1+(dy/dx)2)3/2/|d2y/dx2|
= (1 + 36x2 + 9 + 36x)3/2/6
= (36x2 + 36x + 10)3/2/6
Replace the value x = 1
= (36 (1)2 + 36 (1) + 10)3/2/6
= (36 + 36 + 10)3/2/6
= 742.54/6
= 123.75 units
Read More: Important Questions on Circles
Applications of Radius of Curvature
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The applications of the radius of curvature are as follows:
- It is used in the Cesro equation, which states that a simple curve is an equation that gives the curvature (K) at a point on the curve with arc length (s) from the start of the curve to a given point in relationship sets.
- Also, it is an equation relating the radius of curvature (R) to arc length.
- It can help to find the radius of curvature of the earth along a course in an azimuth.
- The radius of curvature also uses three-part equations for beam bending.
- In addition, it has a specific meaning and signs convention in optical design. All spherical lenses have a center of curvature.
Difference Between Radius and Radius of Curvature
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The difference between radius and radius of curvature can be summed up as follows:
- Radius refers to the distance between the center of a circle or any other point on the circumference of the circle and the surface of the sphere.
- On the other hand, the radius of curvature is defined as the radius of the circle that touches the curve at a given point. It also has the same tangent and curvature at that point.
- Furthermore, the radius has a real figure or shape while the radius of curvature is an imaginary circle.
Read More: Circles Revision Notes
Types of Curvature
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In general, there are two types of curvature, namely outer curvature and inner curvature.
Extrinsic Curvature
- It is a curvature that is a submanifold of a manifold that depends on its particular incorporation.
- Its examples also include the torsion of curves in three-space and curvature.
- It also includes the mean curvature of surfaces in three-space.
Intrinsic Curvature
- It is also a curvature like Gaussian curvature that is apparent to 2-D (two-dimensional) populations of a surface and not just to outside observers.
- Also, it cannot study 3-D (three-dimensional).
Read More:
| Related Topics | ||
|---|---|---|
| Central Angle of Circle | Chord of Circle Length Theorem | Concentric Circles |
| Radius Formula | Area of Segment of a Circle | Tangent Circle Formula |
Things to Remember
- Radius of Curvature refers to the radius of the approximate circle at a specific place.
- Radius of Curvature is defined as the distance between the vertex and the center of curvature.
- It is represented by the symbol R and the curvature vector length.
- The Radius of Curvature Formula is R=(1+(dy/dx)2)3/2/|d2y/dx2|.
- Radius of Curvature is also equal to R = 1/K.
Sample Questions
Ques. Find the radius of curvature of for 3x2 + 2x - 5 at x = 1. (3 Marks)
Ans. We need to find the radius of curvature.
- y = 3x2 + 2x - 5
- dy / dx = 6x + 2
- d2y / dx2 = 6
Using the radius of curvature formula,
R=(1+(dy/dx)2)3/2/|d2y/dx2|
By plotting the values, we get
R = (1+(6x+2)2)3/2 / 6
R=(1+(36x+4 + 24x)3/2 / 6
Now we set x = 1
R=(36 + 5 + 24)3/2 / 6
R = (65)3/2 / 6
R = 87.34
Hence the radius of curvature is 87.34.
Ques. Observe the radius of curvature of 3x3 + 2x - 5 at x = 2. (3 Marks)
Ans. It is given that,
- y = 3x3 + 2x - 5
- dy / dx = 9x2 + 2
- d2y / dx2 = 18x
Using the radius of curvature formula,
we get R=(1+(dy/dx)2)3/2/|d2y/dx2|
Substituting the values we get
R = (1+(814+4 + 36x)3/2 / 18x
Substituting x = 2
We get R = (1296 + 5 + 144)3/2
R = 1413.19
Therefore the radius of curvature is 1413.19 units.
Ques. Find is the radius of curvature for f(x) = 3x3 - 2x - 2 at x = 2. (3 Marks)
Ans. We have,
- y = 3x3 - 2x - 2
- x = 2
- dy/dx = 9x2 - 2
- d2y/dx2 = 18x
Using the radius of curvature formula,
R=(1+(dy/dx)2)3/2/|d2y/dx2|
= (1 + 81x4 + 4 - 36x2)3/2/18x
= (81x4 - 36x2 + 5)3/2/18x
Substitute the value x = 2 .
R = (81(2)4 - 36(2) + 5)3/2/18(2)
= (576 - 72 + 5)3/2/36
= 12305.26/36
= 512.71 units
Ques. Find the radius of curvature for f(x) = 5x3 - 3x2 + x at x = 1. (3 Marks)
Ans. We have y = 5x3 - 3x2 + x and x = 1.
- dy/dx = 15x2 - 6x + 1
- d2y/dx2 = 30x - 6
Using the radius of curvature formula,
R=(1+(dy/dx)2)3/2/|d2y/dx2|
R = (1 + (15x2 -6)2)3/2/(30x -6)
= (1 + 225x4 + 36 -180x2)3/2/(30x -6)
= (225x4 - 180x2 + 37)3/2/ (30x -6)
Substitute the value x = 1.
R = (225(1)4 180(1)2 + 37)3/2/(30 (1) 6)
= (225 - 180 + 37)3/2/24
= 742.54/24
= 30.93 units
Ques. Find the radius of curvature for the curve f(x) = x2. (3 Marks)
Ans. We have the curve, y = x2
- dy/dy=2x
- d2y/dx2=2
Using the radius of curvature formula,
R = (1+(2x)2)3/2/2
= (1+4x2)3/2/2
= (1+4y)3/2/2
Ques. Find the radius of curvature for the curve f(x) = sin x. (3 Marks)
Ans. We have the curve, y = sin x.
- dy/dx = cos x
- d2y/dx2 = -sin x
Using the radius of curvature formula,
R =-(1+(cos x)2)2/3/sin x
=(1+cos2x)3/2/sin x
=(1+cos2x)3/2/y
Ques. Find the radius of curvature for the curve f(x) = ex. (3 Marks)
Ans. We have the curve, y = ex.
- dy/dx = ex
- d2y/dx2=ex
Using the radius of curvature formula,
R = (1+(ex)2)2/3/ex
= (1+e2x)2/3/ex
= (1+y2)3/2/y
Ques. What is the Radius of Curvature Formula? (3 Marks)
Ans. The radius of curvature of a curve y= f(x) at a point is given as
The Radius of Curvature is the reciprocal of the curvature K of the curve at a point. Thus, In case K is the curvature of the curve and R = radius of curvature of the curve, then,
R = 1/K
Ques. Why do we need to measure the curvature and how is it measured? (3 Marks)
Ans. The Curvature tells us how fast the direction is changing as a point moves along a curve. The curvature is usually measured in radians/meters or radians/miles or degrees/miles. The curvature is the reciprocal of the radius of curvature of the curve at a given point or vice versa.
Ques. What is the radius and radius of curvature? (3 Marks)
Ans. The radius of a circle refers to the distance between the center of a circle or any other point on the circumference of the circle and the surface of the sphere. Whereas on the other hand, the radius of curvature refers to the radius of the circle that touches the curve at a given point.
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